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An Abstraction of Whitney's Broken Circuit Theorem

Klaus Dohmen, Martin Trinks

TL;DR

This work generalizes Whitney's broken circuit theorem to sums $\sum_{A\subseteq S} f(A)$ with $f$ valued in an abelian group, introducing a versatile broken-circuit framework and two proofs. The main result yields restricted-sum identities that omit certain dependent configurations, enabling unified derivations of many classical and new results across graphs, hypergraphs, matroids, lattices, and arithmetic functions. Key contributions include applications to hypergraph chromatic polynomials, subgraph component polynomials, domination polynomials, matroid invariants, and new gcd-/lcm-sum expansions for arithmetical functions and $1/\zeta(s)$, plus a convex-geometry generalization. The approach offers a broad, abstract toolkit for inclusion–exclusion-type identities and connects combinatorial structure with arithmetic identities in a unified formalism.

Abstract

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the Möbius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical Möbius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).

An Abstraction of Whitney's Broken Circuit Theorem

TL;DR

This work generalizes Whitney's broken circuit theorem to sums with valued in an abelian group, introducing a versatile broken-circuit framework and two proofs. The main result yields restricted-sum identities that omit certain dependent configurations, enabling unified derivations of many classical and new results across graphs, hypergraphs, matroids, lattices, and arithmetic functions. Key contributions include applications to hypergraph chromatic polynomials, subgraph component polynomials, domination polynomials, matroid invariants, and new gcd-/lcm-sum expansions for arithmetical functions and , plus a convex-geometry generalization. The approach offers a broad, abstract toolkit for inclusion–exclusion-type identities and connects combinatorial structure with arithmetic identities in a unified formalism.

Abstract

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type where is a finite set and is a mapping from the power set of into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the Möbius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical Möbius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).

Paper Structure

This paper contains 16 sections, 2 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Applications
  4. Chromatic polynomial of a hypergraph
  5. Subgraph component polynomial
  6. Domination polynomial
  7. Characteristic polynomial and beta invariant
  8. Maximum-minimums identity
  9. Principle of inclusion-exclusion
  10. Möbius function of a lattice
  11. Arithmetical functions
  12. The classical Möbius function
  13. Euler's totient function
  14. Dirichlet inverse of Euler's totient function
  15. Riemann zeta function
  16. ...and 1 more sections

Key Result

Theorem 1

Let $S$ be a finite linearly ordered set, $\mathscr{C}\subseteq 2^S\setminus\{\emptyset\}$, $\Gamma$ an abelian group (additively written), and $f:2^S\rightarrow \Gamma$ a mapping such that for any $C\in\mathscr{C}$ and $A\supseteq C$, Then, for any $\mathscr{B}\subseteq \{ C\setminus\{\max C\} \mathrel| C\in\mathscr{C} \}$,

Theorems & Definitions (10)

  • Theorem 1
  • proof : First Proof
  • proof : Second Proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4
  • proof : Proof of Theorem \ref{['thm:1']}